In this paper, we study a co-variational inequality problem involving Yosida approximation operators in real uniformly smooth Banach space and define an iterative method to obtain its solution. We prove some properties of Yosida approximation operator for our purpose, that is strongly accretivity and Lipschitz continuity. An existence as well as convergence result is proved for co-variational inequality problem involving Yosida approximation operators using the concept of nonexpansive sunny retraction.
Keywords:
Subject: Computer Science and Mathematics - Applied Mathematics
1. Introduction
Variational inequality theory is an influential unifying methodology for solving many obstacles of pure as well as applied sciences. Hartman and Stampacchia [1] in 1966 initiated the study of variational inequalities while dealing with some problems of mechanics.
The concept of variational inequalities furnish us various devices for modelling many problems existing in variational analysis related to applicable sciences. One can ensure the existence of solution and convergence of iterative sequences using these devices. For applications, see [2,3,4,5,6,7,8,9,10] and references therein.
Alber and Yao [11] first considered and studied co-variational inequalities using nonexpansive sunny retraction concept. They obtained solution of co-variational inequality problem and discussed the convergence criteria. Their work is extended by Ahmad and Irfan [12].
Yosida approximation operators are useful for obtaining solution of various types of differential equations. Petterson [13] first solved the stochastic differential equation by using Yosida approximation operator approach. For the study of heat equations, problem of couple sound and heat flow in compressible fluids, wave equations, etc., the concept of Yosida approximation operator is applicable. For more details, we refer to [14,15,16,17,18].
After above important discussion, the aim of this work is to introduce a different version of co-variational inequality which involve two generalized Yosida approximation operators. We obtain the solution of our problem and we also discuss convergence criteria for the sequences achieved by iterative method.
2. Preface
Throughout this document, we denote real Banach space by and its dual space by . Let be the duality pairing between and . The usual norm on is denoted by , the class of nonempty subsets of by and the class of nonempty compact subsets of by .
The Normalized duality operator is defined by
Some characteristics of normalized duality operator can be discovered in [19].
For the space , modulus of smoothness is given by the function
In fact is uniformly smooth if and only if
The following result is instrumental for our main result.
Proposition 1.
[11] Let be a uniformly smooth Banach space and J be the normalized duality operator. Then, for any , we have
(i)
,
(ii)
, where
Definition 1.
The operator is called
(i)
accretive, if
(ii)
Strongly accretive, if
where is a constant.
(iii)
Lipschitz continuous, if
where is a constant.
(iv)
expansive, if
where is a constant.
Definition 2.
Let be an operator. The operator is said to be
(i)
Lipschitz continuous in the first slot, if
where is a constant.
Similarly, we can obtain Lipschitz continuity of S in other slots.
(ii)
Strongly accretive in the first slot with respect to , if
where is a constant.
Similarly strong accretivity of S in other slots and with respect to other operators can be obtained.
Definition 3.
The operator is called D-Lipschitz continuous if
where is a constant and denotes the Housdörff metric.
Definition 4.
[11] Suppose is the nonempty closed convex subset of . Then an operator is called:
(i)
retraction on , if ,
(ii)
nonexpansive retraction on , if it satisfies the inequality:
(iii)
nonexpansive sunny retraction on , if
for all and for .
Some characteristics of nonexpansive sunny retraction operator are mentioned below, which can be found in [20,21,22].
Proposition 2.
The operator is a nonexpansive sunny retraction, if and only if
for all and .
Proposition 3.
Suppose and is a nonexpansive sunny retraction. Then for all , we have
Definition 5.
The multi-valued operator is called accretive, if
Definition 6.
Let be an operator. The multi-valued operator is said to be -accretive if is accretive and
Definition 7.
We define such that
We call it generalized resolvent operator.
Definition 8.
We define such that
We call it generalized Yosida approximation operator.
Proposition 4.
[23] Let is -strongly accretive and is -accretive multi-valued operator. Then, the operator satisfies the following condition.
That is, is -Lipschitz continuous.
Proposition 5.
If is -strongly accretive, -expansive, -Lipschitz continuous operator and is -Lipschitz continuous, then the operator satisfy the following condition.
where and all the constants involved are positive. That is, is -strongly accretive with respect to the operator .
Proof.
Since , we evaluate
As is expansive and Lipschitz continuous and the generalized resolvent operator is Lipschitz continuous, we obtain
That is,
That is, is -strongly accretive with respect to . □
Proposition 6.
Let be -Lipschitz continuous, -strongly accretive operator and is -Lipschitz continuous, then the operator satisfy the following condition.
where . That is, is -Lipschitz continuous.
Proof.
Since and the generalized resolvent operator are Lipschitz continuous, we obtain
That is
Thus, the operator is -Lipschitz continuous. □
3. Problem Formation and Iterative Method
Suppose is a nonlinear operator, be multi-valued operators and be a multi-valued operator such that is a nonempty, closed and convex set for all . Let be the single-valued operators, be -accretive multi-valued operator and be -accretive multi-valued operator, and be the generalized Yosida approximation operators.
We consider the problem of finding , such that
We call problem (1) as co-variational inequality problem involving generalized Yosida approximation operators..
Clearly for problem (1), it is easily accessible to obtain co-variational inequalities studied by Alber and Yao [11] and Ahmad and Irfan [12].
We provide few characterizations of solution of problem (1).
Theorem 1.
Let be the multi-valued operators, be the nonlinear operator and be a multi-valued operator such that is a nonempty, closed and convex set for all . Let be the single-valued operators, be -accretive multi-valued operator and be -accretive multi-valued operator, and be the generalized Yosida approximation operators, where is a constant. Then the following assertions are similar:
Combining Proposition 3 and Theorem 1, we obtain the theorem mentioned below.
Theorem 2.
Suppose all the conditions of Theorem 1 are fulfilled and additionally , for all , where F is nonempty closed convex subset of and be a nonexpansive sunny retraction. Then and form the solution of problem (1), if and only if
where is a constant.
Using Theorem 2, we construct the following iterative method.
Iterative Method 3.1. For initial , let
Since and are nonempty convex sets, by Nadler[24], there exists and such that
where denotes the Hausdorff metric.
Proceeding in a similar manner, we can find the sequences and by the following method:
for , where is a constant.
4. Convergence Result
Theorem 3.
Suppose be real uniformly smooth Banach space and , for some , is the modulus of smoothness. Suppose F be a closed convex subset of be an operator, be the multi-valued operators, be an operator. Let be a nonexpansive sunny retraction and be a multi-valued operator such that , for all . Let be the multi-valued operators, be operators. Let be the generalized Yosida approximation operator associated with the generalized resolvent operator and be the generalized Yosida approximation operators associated with the generalized resolvent operator . Suppose that the following assertions are satisfied:
(i)
is -strongly accretive with respect to in the first slot, -strongly accretive with respect to in the second slot, -strongly accretive with respect to in the third slot and -Lipschitz continuous in first slot, -Lipschitz continuous in second slot, -Lipschitz continuous in third slot.
(ii)
is -D-Lipschitz continuous, is -D-Lipschitz continuous and is -D-Lipschitz continuous.
(iii)
is -Lipschitz continuous.
(iv)
is -strongly accretive, -expansive and -Lipschitz continuous and is -strongly accretive, -expansive and -Lipschitz continuous.
(v)
is -Lipschitz continuous and is -Lipschitz continuous.
(vi)
is -strongly accretive, -Lipschitz continuous and is -strongly accretive, -Lipschitz continuous.
(vii)
Suppose that
where
Then, there exist and , the solution of problem (1). Also sequences and converge strongly to and , respectively.
Proof.
Using (3) of iterative method Section 3 and nonexpansive retraction property of , we estimate
Applying Proposition 1, we evaluate
Since is -strongly accretive with respect to in the first slot, -strongly accretive with respect to in the second slot, -strongly accretive with respect to in the third slot and applying (ii) of Proposition 1, (8) becomes
As is -Lipschitz continuous in first slot, -Lipschitz continuous in second slot, -Lipschitz continuous in third slot and is -D-Lipschitz continuous, is -D-Lipschitz continuous, is -D-Lipschitz continuous, we have
Using equation (10) and of Proposition 1, we evaluate
In view of the assumption , and clearly is a Cauchy sequence such that we have and . Using (4), (5), (6) of iterative method Section 3, D-Lipschitz continuity of , and the techniques of Ahmad and Irfan [12], it can be shown easily that and are all Cauchy sequences in . Thus, and . Since and are all continuous operators in , we have
It is remaining to show that and . In fact
Hence, and thus . Similarly, it can be shown that and . From Theorem 2, the result follows. □
5. Conclusions
In this work, we consider a different version of co-variational inequalities existing in available literature. We call it co-variational inequality problem which involves two generalized Yosida approximation operators depending on different generalized resolvent operators. Some properties of generalized Yosida approximation operators are proved. Using the concept of nonexpansive sunny retraction, we prove an existence and convergence result for problem (1).
Our results may be used for further generalization and experimental purposes.
Author Contributions
Conceptualization, R.A.; methodology, R.A. and A.K.R.; software, M.I.; validation, R.A., M.I. and H.A.R.; formal analysis, Y.W.; resources, Y.W. and H.A.R.; writing—original draft preparation, M.I.; writing—review and editing, A.K.R.; visualization, A.K.R.; supervision, R.A.; project administration, R.A. and Y.W.; funding acquisition, Y.W. All authors have read and agreed to the published version of the manuscript.
Funding
This research was funded by the National Natural Science Foundation of China (Grant no. 12171435).
Conflicts of Interest
All authors declare that they do not have conflict of interest.
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